Abstract
The present status of the theory of random elastic media is reviewed. A formal solution is derived which gives the tensor of the effective moduli in terms of the correlation functions up to infinite order of the distribution of the local elastic moduli. The formal solution is given in terms of multiple integrals which can be calculated only in favourable situations. Most important is the case of perfect disorder defined by a statistically independent distribution of the elastic moduli. In this case the integrals can be calculated and bounds which are correct to third order are derived. A peculiar difficulty which arises when the method is applied to dynamical problems is discussed.
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References
I.M. Lifshitz and L.N. Rosenzweig, J.Exp.Teor. Fiz.16, 967 (1946).
W.F. Brown J.Chem.Phys 23, 1514 (1955).
V.A. Lomakin J.Appl.Math.Mech 29, 1o48 (1965), translated from Russian.
S.D. Volkov and N.A. Klinskikh Phys.Metals, Metallogr.19, 24 (1965), translated from Russian.
M.J. Beran and J. Molyneux Quart.Appl.Math.24, 1o7 (1966).
E. Kröner J.Mech.Phys.Solids 15, 319 (1967).
C. Eimer Arch.Mech.Stos 19, 4 (1967).
V.V. Bolotin and V.N.Moskalenko Sov.Phys.Dokl.13, 73 (1968).
M.J.Beran and J.J. McCoy Int.J.Solids Struct.6, lo35 (197o)
V.V. Novoshilov Prikl.Mat.Mech 34, 67 (197o).
W.M. Lewin Prikl.Mat.Mech 35, 744 (1971).
P. Mazilu Rev.Roum.Math.Pur.Appl 17, 261 (1972).
R. Zeller and P.H. Dederichs Phys.stat.sol(b)55, 831 (1973). —
P.H. Dederichs and R. Zeller Z.Phys 259, 103 (1973).
E. Kröner, I nt.J.Eng.Sci 11, 171 (1973).
J. Korringa J.Math.Phys 14, 509 (1973).
M. Hori J.Math.Phys 14, 514, 1942 (1973).
M. Hori and F. Yonezawa J.Math.Phys 15 (1974), in press.
E. Kröner Statistical Continuum Mechanics, CISM Courses and Lectures No. 92, Udine 1971, Springer- Verlag Wien-New York, 1972.
A.G. Fokin and T.D. Shermergor, Transl. from PMM 32, 66o (1968). —
E. Kröner Internal stresses in crystals and in the earth, this symposium.
H.J. Bunge, Mathematische Methoden der Textur analyse, Akademie-Verlag, Berlin, 1969.
A.V. Hershey J.Appl.Mech 21, 236 (1954)•
E. Kroner Z.Phys 151, 504 (1958).
G. Kneer phys.stat.sol 9, 825 (1965).
P.R. Morris Int.J.Eng.Sci 8, 49 (197o).
J.W. Hutchinson Proc.Roy.Soc A 319, 247 (197o)
J.W.F. Bishop and R. Hill, Phil.Mag.42, 414, 1298 (1951).
R. Hill Proc.Phys.Soc A 65, 349 (1952).
M.S. Howe Proc. Roy. Soc A 31, 479 (1973)
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© 1974 D. Reidel Publishing Company, Dordrecht-Holland
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Kröner, E. (1974). Statistical Problems in the Theory of Elasticity. In: Thoft-Christensen, P. (eds) Continuum Mechanics Aspects of Geodynamics and Rock Fracture Mechanics. NATO Advanced Study Institutes Series, vol 12. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2268-2_9
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DOI: https://doi.org/10.1007/978-94-010-2268-2_9
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